Question

Prove that 3+√5 isirrational

Answer 1

Step-by-step explanation:

let us assume , to the contrary ,that

$3+ \sqrt{5} \: \: \bold{is \: rational \: }$

then there exist co-prime a and b

$(b \: \not = 0) \: \: \bold{such \: that} \\ (3 + \sqrt{5} ) = \frac{a}{b} \\ \sqrt{5} = \frac{a}{b} - 3 \\ \sqrt{5} = \frac{a - 3b}{b}$

since a and b are integers,so

$\frac{a - 3b}{b}$

is rational

$\bold{thus \: \sqrt{5} \: \: is \: also \: rational \: }$

but this contradict the fact that

$\sqrt{5}$

is rational .so our assumption is incorrect

$hence \: 3 + \sqrt{5} \: \: is \: irrational \:$